Hence, provided that \(\eta \)  is sufficiently small, it follows from the \(C^1\)  linearization result given in [1]} that there exists a \(C^1\) -diffeomorphism \(\Phi \colon Y_\infty \rightarrow Y_\infty \)  such that
\(\Phi \circ F={B}^*\circ \Phi .\) 
